1913] ON THE MOTION OF A VISCOUS FLUID 195
BD-/v> 177 the regular motion is necessarily unstable. As the fluid moves under the laws of dynamics, the initial increase of certain disturbances may after a time be exchanged for a decrease, and this decrease may be without limit.
At the other extreme when v is very small, observation shows that the tangential traction on the walls, moving (say) with velocities + £7, tends to a statistical uniformity and to become proportional, no longer to £7, but to U2. If we assume this law to be absolute in the region of high velocity, the principle of dynamical similarity leads to rather remarkable conclusions. For the tangential traction, having the dimensions of a pressure, must in general be of the form
............................... (3r)
D being the distance between the walls, and f an arbitrary function. In the regular motion (z large) f(z) — Zz, and (37) is proportional to U. If (37) is proportional to U2, f must be a constant and the traction becomes independent not only of /*, but also of D.
If the velocity be not quite so great as to reduce / to constancy, we may take
where a and b are numerical constants, so that (37) becomes
apU' + bnUfD ............................... (38)
It could not be assumed without further proof that & has the value (2) appropriate to a large z\ nevertheless, Korteweg's equation (6) suggests that such may be the case.
From data given by Couette I calculate that in c.G.S. measure
a = -000027.
The tangential traction is thus about a twenty thousandth part of the pressure (\pUz) due to the normal impact of the fluid moving with velocity U,
Even in cases where the steady motion of a viscous fluid satisfying the dynamical equations is certainly unstable, there is a distinction to be attended to which is not without importance. It may be a question of the time during which the fluid remains in an unstable condition. When fluid moves between two coaxal cylinders, the instability has an indefinite time in which to develop itself. But it is otherwise in many important problems. Suppose that fluid has to move through a narrow place, being guided for example by hyperbolic surfaces, either in two dimensions, or in three with symmetry about an axis. If the walls have suitable tangential velocities, the motion
13—2ergy T' of the motion (1) which is the difference of these two, and so makes the velocities vanish at the boundary. The motion M' with velocities u', v', w' does not in general satisfy the dynamical equations. We have